CSE 599d - Quantum Computing When Quantum Computers Fall Apart

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1 CSE 599d - Quantum Comuting When Quantum Comuters Fall Aart Dave Bacon Deartment of Comuter Science & Engineering, University of Washington In this lecture we are going to begin discussing what haens to quantum comuters when we move away from the closed system view of quantum theory and encounter the reality that real quantum system are oen quantum systems. Oen quantum systems interact with their environment and decohere. When we try to control real quantum systems, we aren t able to erfectly control them. When we try to erform rojective measurements on a real quantum systems, we don t erform exactly erfect rojective measurements. When we try to reare real quantum systems into a articular state, we don t succeed in rearing this state with erfect certainty. All of these issues must be addressed if we are really going to ( build a quantum comuter, ( accet that quantum comutation is a valid model deserving of the moniker digital comuter. I. QUANTUM NOISE Suose that our hay qubit ψ is sitting there minding its business, when the cold hard reality that is not alone in the universe shows its head. In articular another qubit, in the state + comes along and interacts with our qubit by a controlled-not, controlled from the extra qubit. What will the effect of this evolution be on our qubit? Well we can calculate the Kraus oerators: A = B C X ( B + B = I The evolution of our initial state ρ = ψ ψ is then given by A = B C X ( B + B = X ( ρ A ρa + A ρa = ( ρ + XρX. ( We can interret this evolution as describing the rocedure of doing nothing to our qubit with robability and with robability alying the unitary oerator X to our qubit. This is an examle of quantum noise on our qubit. But notice that this quantum noise is very similar to what we might call classical noise in the comutational basis. In the comutational basis, this reresents nothing more that fliing the bit. Now suose that we relace the controlled-not in this oeration with a controlled-u oeration. Then we can similarly calculate that A = B C X ( B + B = I A = B C X ( B + B = U (3 The evolution of this sueroerator can be interreted as doing nothing with robability and alying U with robability. Now if U is the Z gate, then, this evolution does something kind of strange to our qubit, when exressed in the comutational basis. In articular the Z gate does not change the amlitude of a state in the comutational basis: α + β α β. Thus if we are talking about measurements in the comutational basis the effect of this noise does not change the robabilities of the two outcomes. But if we examine the density matrix, then something has haened to our system: αβ α β β ( αβ α β β + [ αβ α β β = ( αβ Thus we see that the off diagonal elements of the the density matrix have decayed. If, for examle =, then the density matrix has gone from a coherent mixture to an incoherent mixture. This is a articulary nasty form of quantum noise, because if, say and are energy eigenstates, then the energy of the qubit has not changed, but the system has lost some of it s coherent roerties. Now what is bad about the quantum noise we have described above? Well the real roblem is that it is not reversible. Suose that the interactions we describe above haen, but then the extra qubit with with our qubit (4

2 interacted becomes inaccessible to us. Well then can we restore our qubit back to it s original state? The answer is no! Why? Well one way to see this to examine the action of the noise rocesses which can occur to a qubit on the Bloch ball. A rocess like our first noise rocess, roduces the evolution (I + n xx + n y Y + n z Z (I + n xx + n y Y + n z Z + (I + n xx n y Y n z Z = (I + n xx ( (n y Y + n z Z. (5 Thus we see that the states in the Bloch ball are shrunk around the Y and Z directions (into a Bloch ellisoid. If we are to reverse this shrinking, there must be a rocess which reverses this shrinking. But no such valid oerator sum reresentation evolution can roduce this. Why? Because it would need to take states with n < to states with n =. But sueroerators are linear, $[am + bn = a$[m + b$[n. So if the there exists a sueroerator which erforms this for our above oeration, it must take (I ±( Y (I ±Y. This imlies ($[I+( $[Y = (I + Y and ($[I ( $[Y = (I Y, or ( $[Y = Y and $[I = I. So alying this to sueroerator to (I + Y we obtain (I + Y which is not a valid density matrix from >. Thus, since sueroerators ma density matrices to density matrices, we have obtained a contradiction, and there must exist no such sueroerator which erforms this ma. Thus we see that the above rocess we have described is irreversible, given that we throw away the system with which our system has interact to roduce the original sueroerator. Let s describe a few of the more imortant mechanism which we encounter in studying quantum systems. In articular we will focus on the action of these quantum noise rocesses on a single qubit. A. Deolarizing Channel In a deolarizing channel, with robability the qubit is left untouched and with robability the qubit is fully mixed. Fully mixed? This means that (I + n σ I. How can we achieve this rocess? One way is to note that XρX + Y ρy + ZρZ = 3 I (6 To seen this note that XXX = X, XY X = X, and XZX = Z so that if ρ = (I + n xx + n y Y + n z Z, then XρX = (I + n xx n y Y n z Z. Similarly Y ρy = (I n xx + n y Y n z Z and ZρZ = (I n xx n y Y + n z Z. Putting these together yeilds the exression. Thus we see that we can roduce the deolarizing channel if we have the four Kraus oerators A = I, A = 3 X A = 3 Y A 3 = 3 Z (7 Then the evolution of ρ will be ρ ( ρ + I (8 What does this do on the Bloch ball? Well it simly shrinks the ball along all directs equally: n ( n (9 B. Phase Daming Channel Suose that our qubit is sitting there, minding its business. The environment, however is always out there and always lanning devious business for our qubit. Suose that the environment is in a state but every once in a while, it scatters off of the qubit. In articular this scattering may deend on the state of the qubit, i.e. if the qubit is in the state, then there may be an amlitude that the environment scatters into the state and if the qubit is in the state, there may be a different amlitude that the environment scatters into the state. Consider, for examle, the situation where the magnitude of the evolution is the same: ( + ( + (

3 The Kraus oerators for this rocess can be calculated, assuming the environment starts in the state, and is given by [ [ A = A = A = ( What does this evolution do to an arbitrary inut density matrix? It is easy to see that it only effects the off diagonal matrix elements: [ αβ α β β ( αβ α β β + β = ( αβ ( α β β ( But notice that this is the same as the evolution we described by couling a controlled-z to our system. But that evolution had different Kraus oerators. That s kind of interesting. In fact these two evolutions, which are call hase daming, reresent (u to that scaling the same sueroerator. There is a degree of freedom in our sueroerators which we haven t discussed, and we will now remedy that situation 3. Unitary Freedom of the Oerator Sum Reresentation To understand the freedom in the oerator sum reresentation Kraus oerators, it is useful to first understand a very cool way to fully characterize a sueroerator. Suose we have some sueroerator $ with Kraus oerator A i. Now one way to characterize this sueroerator is to check how it oerators on different states and, given enough of these states, one can regain all of the information describing the sueroerator. This is known as rocess tomograhy, and we won t discuss it here. But there is a much cooler way to secify a sueroerator. Let our system have a Hilbert sace H A. Attach an ancilla system of the same dimension as our Hilbert sace to H A so our full Hilbert sace is H = H A H B. Now define the (unnormalized state on this joint system ψ AB = i A i B (3 where d = dimh A. This is the maximally entangled states (if we had roerly normalized it shared between H A and H B. Now the cool thing about ψ is that it allows us to secify states on subsystem A using states on subsystem B. In articular note that is we let φ B = d a i i B, then φ B ψ AB = a i i A = φ B. (4 Evidentally, ψ AB can serve as a ma from states on B to states on A. What consequence does this have for sueroerators? Well define the oerator τ = ($ I[ ψ AB ψ AB (5 Now what is cool is that τ gives us all of the information about $ that we need to reconstruct $! By looking at how a sueroerator acts on half of a maximally entangled quantum state, we can comletely characterize the sueroerator. How do we see this? Well again defining φ B = d a i i B, we can calculate φ B τ φ B φ B τ φ B = φ B ($ I[ ψ AB ψ AB φ B = ($[ φ B ψ AB ψ AB φ B = $[ φ A φ A (6 Suose that we decomose τ as τ = k v k AB v k AB. Then define ma A k ( φ A = φ B v k AB. This ma is certainly linear, so we can just think about it as a linear oerator. But then we have A k φ A φ A A k = φ B v k AB v k AB φ B = φ B τ φ B = $[ φ A φ A (7 k k Thus we have seen that by seeing how $ acts on half of a maximally entangled state, we can calculate τ, and using τ we can then find the oerator sum reresentation of $. Okay now that we ve got that cool result out of the way, we can discuss the freedom in the oerator sum reresentation. Suose that we have two sueroerators with Kraus oerator {A k } and {B k }. Assume that they have the

4 same number of oerators and if they do not, then just ad on of the sets with zero oerators. Then we will show that if these reresent the same oerator then these reresent the same sueroerator iff there exists a unitary matrix with entries u mn such that 4 A k = l u kl B l (8 Again attach an ancilla sace B with the same dimension as A. Then define the states resulting from acting with A k or B l on one half of (not roerly normalized maximally entangled state between A and B. a k = b l = (A k i A i B (B l i A i B (9 Then it must be that the oerator which was τ above, reresenting the result of acting with I $ on the maximally entangled state, are, for these two sueroerators, τ = k a k a k and σ l = l b l b l. But these are (u to that normalization valid density matrices. The only way these can be the same density matrices and hence reresent the same sueroerator is, from a few lectures back, if a k = l u kl b l ( for some unitary matrix with entries u kl. Now for arbitrary states ψ, we find that A k ψ A = ψ B a k = l ψ B u kl b l = l u kl B l ψ ( Since this must hold for arbitrary ψ this imlies that A k = l u kl B l ( as desired. Conversely, suose that A k = l u klb l. Then k A kρa k = k l u klb l ρ m u km B m. But k u klu km = δ lm, so this imlies that k A kρa k = l B lρb l. Thus we have shown that two sueroerators are the same iff the sets of Kraus oerators for these sueroerators can be related to each other by a unitary transform over the different oerators (not on the oerators themselves. Let s see this for the two channels where we first noted this. In the first sueroerator the Krauss oerators were A = I A = Z (3 and in the second case the Kraus oerators were [ B = B = [ B = (where we have rescaled the first to make this corresondence correct. We can rewrite these second Kraus oerators as B = I B = (I + Z B = (I Z (5 Then it the question becomes whether we can find a unitary which transforms between these two sets o matrices. Indeed we can, and it is given by U = ( ( ( (4 (6 How did I find this? Well I looked at the coefficients in from of the I and Z searately and derived equations from these conditions. Of course the answer I got is not unique: the final row can have an arbitrary hase, for examle.

5 5 C. Amlitude Daming Channel A final, very interesting channel is the amlitude daming channel. This channel is used to describe, say the decay of an a two level system. Suose that our environment is initially in the state E reresenting the vacuum. Then if the system is in its lower energy state, which we take to be, then nothing haens. However if the system is in its higher energy state, which we take to be the state, then there is an amlitude for the system to relax to the ground state and emit a hoton. The unitary descrition of this rocess is E E E E + E (7 Calculating the Kraus oerators we find that they are given by A = A = [ (8 Notice that A is not a hermitian oerator (as in our revious Kraus oerators. How does a density matrix evolve under this evolution? Well we can calculate that [ αβ α β β αβ [ α β β β + = ( + β αβ ( α β β (9 ( Thus we see that there is oulation transfer between the and state, along with an daming of the off diagonal elements of the density matrix. If reresents the amlitude for this to occur er unit time ste, for a time t, then the oulation of the higher energy level state decays like ( t t t ex( t. So this is just the exonential decay of a higher energy state to a lower energy state. Notice also that the amlitude daming channel does not send I to I. Quantum noise which reserve the maximally mixed state (like I for a qubit are called unital. Thus amlitude daming is not a unital sueroerator. D. Quantum Noise We ve seen some ossible quantum noise rocesses in this lecture. The question we will worry about next is how we might ossibly overcome this quantum noise. One articular worry is that the noise is secified by Kraus oerator {A k } and these form a continuous set of oerators. Of course, if we really think about classical noise, then it will also form a continuous set. We aren t worried there because we can think about noise in the classical case as deterministic rocedures occurring with robabilities. Are their equivalent notions for quantum theory? Stay tuned and you will find out!